Homoclinic orbits near saddle-center fixed points of Hamiltonian systems with two degrees of freedom

Patrick Bernard (1) , Clodoaldo Grotta Ragazzo (2) , Pedro Salamao (2)

1 IF - Institut Fourier
2 Instituto de Matematica e Estatistica

We study a class of Hamiltonian systems on a 4 dimensional symplectic manifold which have a saddle-center fixed point and satisfy the following property: All the periodic orbits in the center manifold of the fixed point have an orbit homoclinic to them, although the fixed point itself does not. In addition, we prove that these systems have a chaotic behavior in the neighborhood of the energy shell of the fixed point.

On étudie une classe de systèmes Hamiltoniens, sur une variété symplectique de dimension 4, admettant un point fixe de type selle-centre. On montre que ces systèmes ont la propriété suivante: Chaque orbite périodique de la variété centrale du point fixe a une orbite homocline, mais le point fixe lui-même n’a pas necessairement d’orbite homocline. On montre de plus que ces systèmes ont un comportement chaotique au voisinage de la surface d’énergie du point fixe.

Damos exemplos de sistemas Hamiltonianos sobre uma variedade simplética de dimensao 4, com um ponto fixo sela-centro, apresentando uma interessante propriedade: Toda orbita periodica na variedade central do ponto fixo tem uma orbita homoclinica a ela, apesar de o ponto fixo nao ter. Alem disso, provamos que tais sistemas tem um comportamento caotico na vizinhancao nivel de energia do ponto fixo.

Domaines: Systèmes dynamiques [math.DS]

Patrick Bernard, Clodoaldo Grotta Ragazzo, Pedro Salamao. Homoclinic orbits near saddle-center fixed points of Hamiltonian systems with two degrees of freedom. Asterisque, 2003, 286, pp.151-165. ⟨hal-01251205⟩